Let X denotes the number of times heads occur in n tosses of a fair coin. If P(X=4) , P(X=5) and P(X=6) are in AP, then the value of n is

Step-by-step solution

Clearly, X is a binomial variate with parametres n and p= 1 2 such that aligned P(X=I) & = ^n C_r p^r q^ n-r = ^n C_r ( 1 2 )^r ( 1 2 )^ n-r & = ^n C_r ( 1 2 )^n aligned Now, P(X=4), P(X=5) and P(X=6) are in AP. array lr & 2 P(X=5)=P(X=4)+P(X=6) & 2 ^n C₅ ( 1 2 )^n= ^n C₄ ( 1 2 )^n+ ^n C₆ ( 1 2 )^n & 2 ^n C₅= ^n C₄+ ^n C₆ & 2 n! (n-5)!5! = n! (n-4)!4! + n! (n-6)!6! & 2 5(n-5) = 1 (n-4)(n-5) + 1 6 5 & n^2-21 n+98=0 & (n-7)(n-14)=0 & n=7 or 14 array